Linking the TPR1, DPR1 and arrow-head matrix structures. (English) Zbl 1132.15009
Summary: Some recent polynomial root-finders rely on effective solution of the eigenproblem for special matrices such as DPR1 (that is, diagonal plus rank-one) and arrow-head matrices. We examine the correlation between these two classes and their links to the Frobenius companion matrix, and we show a Gauss similarity transform of a TPR1 (that is, triangular plus rank-one) matrix into DPR1 and arrow-head matrices. Theoretically, the known unitary similarity transforms of a general matrix into a block triangular matrix with TPR1 diagonal blocks enable the extension of the cited effective eigen-solvers from DPR1 and arrow-head matrices to general matrices. Practically, however, the numerical stability problems with these transforms may limit their value to some special classes of input matrices.
MSC:
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 15B57 | Hermitian, skew-Hermitian, and related matrices |
| 65H05 | Numerical computation of solutions to single equations |
| 12Y05 | Computational aspects of field theory and polynomials (MSC2010) |
| 26C10 | Real polynomials: location of zeros |
| 65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |
Keywords:
algebraic eigenproblem; similarity transforms; arrow-head matrices; diagonal plus rank-one matrix; triangular plus rank-one matrix; polynomial root-finders; Frobenius companion matrix; numerical stabilityReferences:
| [1] | Wilkinson, J. H., The Algebraic Eigenvalue Problem (1965), Clarendon Press: Clarendon Press Oxford · Zbl 0258.65037 |
| [2] | Golub, G. H.; Van Loan, C. F., Matrix Computations (1996), The Johns Hopkins University Press: The Johns Hopkins University Press Baltimore, MD · Zbl 0865.65009 |
| [3] | Stewart, G. W., Matrix Algorithms, Volume II: Eigensystems (1998), SIAM: SIAM Philadelphia, PA · Zbl 0910.65012 |
| [4] | D.A. Bini, L. Gemignani and V.Y. Pan, QR-like algorithms for generalized semiseparable matrices, Technical Report 1470, Department of Math., University of Pisa, Pisa, Italy, July 2003.; D.A. Bini, L. Gemignani and V.Y. Pan, QR-like algorithms for generalized semiseparable matrices, Technical Report 1470, Department of Math., University of Pisa, Pisa, Italy, July 2003. |
| [5] | Bini, D. A.; Gemignani, L.; Pan, V. Y., Inverse power and Durand/Kerner iteration for univariate polynomial Root-finding, Computers and Mathematics with Applications, 47, 2/3, 447-459 (2004) · Zbl 1054.65046 |
| [6] | Bini, D. A.; Gemignani, L.; Pan, V. Y., Improved initialization of the accelerated and robust QR-like polynomial root-finding, Electronic Transactions on Numerical Analysis, 17, 195-205 (2004) · Zbl 1065.65065 |
| [7] | V.Y. Pan, B. Murphy, R.E. Rosholt, Y. Tang and X. Yan, Eigen-solving via small-rank modification (to appear).; V.Y. Pan, B. Murphy, R.E. Rosholt, Y. Tang and X. Yan, Eigen-solving via small-rank modification (to appear). |
| [8] | Stewart, G. W., Matrix Algorithms, Volume II: Eigensystems (1998), SIAM: SIAM Philadelpia, PA · Zbl 0910.65012 |
| [9] | Vanderbil, R.; Van Camp, E.; Van Barel, M.; Mastronardi, N., Orthogonal similarity transformation of a symmetric matrix into a diagonal-plus-semiseparable one with free choice of the diagonal, Numerische Mathematik, 102, 709-726 (2006) · Zbl 1086.65040 |
| [10] | Eidelman, Y.; Gohberg, I., Fast inversion algorithms for a class of block structured matrices, Contemporary Mathematics, 281, 17-38 (2001) · Zbl 1004.65038 |
| [11] | Vanderbil, R.; Van Barel, M.; Mastronardi, N., A note on the representation and definition of semiseparable matrices, Numerical Linear Algebra with Applications, 12, 839-858 (2005) · Zbl 1164.15341 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
